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Metamath Proof Explorer

Theorem cjrebd

Description: A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cjrebd.2	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) = 𝐴 )
Assertion	cjrebd	⊢ ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjrebd.2	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) = 𝐴 )
3		cjreb	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ∗ ‘ 𝐴 ) = 𝐴 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ↔ ( ∗ ‘ 𝐴 ) = 𝐴 ) )
5	2 4	mpbird	⊢ ( 𝜑 → 𝐴 ∈ ℝ )